EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no longer relevant".

As some of you may already know, there are plans in the making for a Museum of Mathematics in New York City. Some of you may have already seen the Math Midway, a preview of the coming attractions at MoMath.

I've been involved in a small way, having an account at the Math Factory where I have made some suggestions for exhibits. It occurred to me that it would be a good idea to solicit exhibit ideas from a wider community of mathematicians.

What would you like to see at MoMath?

There are already a lot of suggestions at the above Math Factory site; however, you need an account to view the details. But never mind that; you should not hesitate to suggest something here even if you suspect that it has already been suggested by someone at the Math Factory, because part of the value of MO is that the voting system allows us to estimate the level of enthusiasm for various ideas.

Let me also mention that exhibit ideas showing the connections between mathematics and other fields are particularly welcome, particularly if the connection is not well-known or obvious.

A couple of the answers are announcements which may be better seen if they are included in the question.

Maria Droujkova: We are going to host an open online event with Cindy Lawrence, one of the organizers of MoMath, in the Math Future series. On January 12th 2011, at 9:30pm ET, follow this link to join the live session using Elluminate.

George Hart: ...we at MoMath are looking for all kinds of input. If you’re at the Joint Math Meetings this week, come to our booth in the exhibit hall to meet us, learn more, and give us your ideas.

I'm reminded of the following quote, which perhaps would be good to include in the museum: "Numbers exist only in our minds. There is no physical entity that is the number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe." - Linear Algebra by Fraleigh + Beauregard
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Zev ChonolesDec 25 '10 at 20:38

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What an opportunity! Clearly, the fact that many of us mathematicians ourselves don't even know about this project (or related ones mentioned in other responses) means, above all, we need to hire marketing professionals! And designers should build the exhibits. (But as for content, I've always liked the Borromean rings: en.wikipedia.org/wiki/Borromean_rings)
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Eric ZaslowDec 26 '10 at 3:09

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I'm wary of both marketing professionals and designers. We are interested neither in selling junk people do not really need, nor in trying to beautify something that is ugly by its nature. If anything, we should get a few high level math. people with good taste and some knowledge of the outside world to make decisions about what to do. But I doubt it'll be done. I bet Percy Diaconis, say, has been neither invited as a consultant, nor even told of the project.
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fedjaDec 26 '10 at 15:34

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@fedja : While Diaconis doesn't seem to be involved, the advisory board (listed here : momath.org/about/advisory-council) includes a lot of very good mathematicians, for example Bjorn Poonen. That being said, I'm still pretty skeptical that a "museum of mathematics" is possible...
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Andy PutmanDec 26 '10 at 20:34

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@Timothy : My skepticism comes precisely from the math sections of a number of science museums I have been too. They've all been pretty lame (and that's not just Andy the math-snob talking -- my wife and kids haven't enjoyed them either). We just don't have cool things like robots or spaceships or dinosaur bones or life-size models of the human heart to show off!
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Andy PutmanDec 26 '10 at 23:18

90 Answers
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Build a fundamental region of a Platonic solid out of mirrors facing inward, e.g., $1/48$ of a cube, omitting the side of the tetrahedron which is part of the exterior of the solid. When you look into those three mirrors, you see copies of yourself looking into a Platonic solid from each of the other fundamental regions.

If you truncate the vertex corresponding to the center of the regular polyhedron appropriately with an opaque triangle, the mirror images of the triangle form the polyhedron or the dual. I think a few of these, made by another math major in my year, might still be in the math lounge at New College.

This is a striking visual effect which can be observed by nonmathematicians in passing. Similarly, two large vertical mirrors set at an angle of $\pi/n$ show the viewer as one of $2n$ copies.

I don't believe that watching working sessions of mathematicians, even with commentary, would be particularly inspiring or interesting to non-mathematicians. What we do is far too foreign. Why would they want to watch us struggle through something they don't understand and have no a priori interest in?
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Deane YangJan 9 '11 at 0:56

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How about movies of kids solving math (or other practical geometry) problems cooperatively, in classrooms even. If you were a kid in a boring school, you might be very gratified to see how a good problem solving session in school might operate. If it were done in the math circle fashion, kids could be motivated to join something like them. They could be arranged by grade level, or you could choose easier or harder ones. Grown up mathematicians would only be one of a series. Come to think of it, math circle organizing could be a major activity of the museum, like glee clubs
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sigoldberg1Jan 11 '11 at 11:00

A slide rule! The physical embodiment of the isomorphism between $\left(\mathbb{R},\cdot\right)$ and $\left(\mathbb{R},+\right)$. There are pretty pieces of history here too - Napier's bones and so forth. A giant one (maybe >1m long) mounted on a wall so that people can make it work - now that would plant the idea of the isomorphism making + and * "the same" operation deeply in the mind of anyone who played around with it seriously.

Heartily second the slide rule but a wide variety showing the many forms that these were implemented. The linear, spiral, cylindrical to generate scales multi feet long for five figure accuracy. Especially relevant in 2014 as the 400th anniversary of Napier's publication of logarithms in 1614.
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David WalkerJan 22 '14 at 18:01

I would enjoy a "hall of infinities", listing countable ordinals... not all of them, but enough to get the idea across. It's possible to draw nice pictures of some them, at least up to $\omega^3$ or so, and even kids know how to count, so they might enjoy knowing what comes after the numbers they learned about in school.

I tried to present this information in story form in "week236" of This Week's Finds.

Actually, now that I think about it, there should be a "hall of numbers" that starts by listing lots of interesting natural numbers and then moves on to countable ordinals.

MIT has an "infinite corridor" that would do well for this, but I guess a shorter version would still be okay.

A wonderful interactive mathematics exhibition called Beyond Numbers was designed by the Maryland Science Center and the George Washington University Department of Mathematics, especially the co-director Rodica Simion. See http://www.gjbgraphics.com/usefulstring/BNTofC.html. It was displayed during the period 1994-1999. Many of the ideas for this exhibition could be carried over to MoMath.

I would like to see RSA encryption included somehow, ideally in a hands-on way (letting them do some arithmetic with aid of calculators which are part of the exhibit) so that people can get the sense that whenever there is an https:// in their browser, a lot of simple but remarkable arithmetic is happening in the background.

If you want to give the audience some sense for what mathematical argument is about, I like the topic of divisibility rules (by 2, 3, 4, 9, etc). Most people have seen these but take them completely for granted - indeed, some people take "ends in an even digit" as a suitable definition for even number. One main characteristic which separates mathematicians from the rest of the world is seeing such a rule and asking "does that always work, and if so why?" So perhaps one could first put some plausible false rules out there to create some doubt and the arguments that these rules work - both with algebra and if possible avoiding algebra. I found that emphasizing this material worked well in a class I taught for future elementary school teachers. I told them that even most/ all of their science major friends who passed AP calculus didn't really know why these rules work, so they had learned something special.

A bubble table, like the one at the Exploratorium. I couldn't quickly find a good link at the Exploratorium website, so check out the list of Google images instead. It would be particularly cool to connect this with a discussion of minimal surfaces.

Edit: Just to give more details -- The bubble table at the Exploratorium is a large (4 feet?) and shallow (4 inches?) bath filled with bubble solution, at waist height of a 6 year old. The museum provides metal loops which visitors use to make large tubular bubbles. It is particularly amazing to lift the hoop up and the pull it down over your head: you get a moment of looking out of a bubble.

I don't remember if they provide other wire frames. It would be cool to have the standard ones to play with (one-skeleta of Platonic solids) and various saddle inducing frames (say, subgraphs of the one-skeleton of the cube). Also interesting: wire frames of knots (interesting unknots, trefoil, figure eight) shaped to allow seeing their Seifert surfaces. Another suggestion: parallel plates of clear plastic connected by rods, to allow the creation of Steiner networks (or at least their approximation).

A few years ago there was an exhibition devoted to mathematics which took place at the Science Museum near the Hebrew University of Jerusalem and also at the Abu Dis Al Kuds University. This was an Italian-Isreali-Palestinian joint endeavor. There were many exhibits (and some were mentioned already among the answers) like: The decimal number system, exponential growth, Konisberg bridges, Tilings periodic and non periodic, knots, The Tower of Hanoi Game, Soap bubbles, Reuleaux triangle, models for graphs of polyhedra, demonstration of Buffon's needle problem, and many more. Some movies (in Hebrew, but still easy to understand) can be found here http://www.cet.ac.il/math/mada.asp See also here

Dear Patrick, Yes it is in Hebrew, (but you can see what the exhibits are and guess what is said). In any case I think the mathematics museum exist now as a permanent exhibition in Abu Dis University. If I will fine more material/pictures I will add them.
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Gil KalaiDec 31 '10 at 11:02

Some mathematics was motivated by astronomy in ways which are hard to notice now due to light pollution and alternatives to staring at the sky at night. I would like to see an exhibit which shows the motions of the planets, Sun, and Moon, sped up and made easier to see, along with a presentation of mathematical results and techniques developed for astronomy, from numerical methods to mechanics to Kepler's laws. Newton and Euler contributed extensively to the mathematics of astronomy, and astronomy influenced many of their mathematical works.

I come across "mind reading" games based on elementary number theory from time to time; e.g. http://www.digicc.com/fido/. It bugs me a bit when people are wowed by such tricks, but not enough to sit down and figure out the mechanics of the thing. But the surprise factor may make a good museum activity -- where the second part of the activity is teaching why the trick works the way it does.

I'd love to see large and detailed historical montages centered around specific developments or results that took significant time and evolution from conception or conjecture to actual proof. For example, we could see a large montage of the development of the proof of Fermat's theorum from Fermat's cryptic anecdote through 2 centuries of developments in number theory,algebra and elliptic curve theory concluding with Wiles' proof of the Taniyama–Shimura conjecture for semistable elliptic curves and Ribet's proof of the epsilon conjecture.

The level of detail could be modular-several levels of explaination could be present from general audience to PHD level.

I'd love to see an exhibit devoted to beautiful and intuitive proofs. Most of us mere mortals will never be able to understand Wiles' proof of Fermat's Last Theorem, but there are some phenomenally interesting and important proofs out there that the average person might be excited to learn about. For instance, using Cantor Diagonalization to prove the uncountability of real numbers. Fascinating and accessible!

A history of Pythagoras' theorem - from Egypt and Babylon through to the proof, then higher dimensional versions, and then a jump from that to non-Euclidean geometry (surfaces of positive and negative curvature), and then introducing the idea of a metric space, with $\mathbb{R}^2$ as an example.

An exhibit on the role of computers in pure (and applied) mathematics. It would especially be nice to see something about experimental mathematics and viewing math, at times, as not purely deductive, but even empirical. I think this would give an idea as to how some mathematicians work and think, and also emphasize the growing importance of computers in verifying or finding new theorems.

I don't think enough people know about the deductive part of mathematics. Too many people believe calculations and mathematics are the same thing. An exhibit on experimental mathematics could help, but this would have to be done carefully.
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Douglas ZareDec 31 '10 at 10:26

How about leading them through an interesting problem, like a geometry IMO problem or, if that is asking too much, a Mathcounts problem? It could be set up on square tiles, the left most of which would contain the problem, with the following tiles showing the steps of the solution. It should be a problem that can be written such that viewers see a surprise toward the end, thereby possibly giving a glimpse into why mathematicians enjoy so much what they do. Although a Mathcounts problem would no doubt be accessible, a very beautiful IMO problem could be inspiring. Very likely, it would be entertaining for both children and their parents.

One might also include multiple solutions to a problem to dispel the notion that for each problem only one solution exists.

One can see examples of interesting presentations and ideas for problems at Rusczyk's Mathcounts channel at

An exhibit - with both individual workstations and one whose results are projected (to draw people in) - where users use parallelograms to specify affine transformations which in turn define Iterated Function Systems and their associated fractals. See here for background. With a few moves/ clicks a user can start making fractal ferns, clouds, spirals and starfish as well as classical objects such as the Sierpinski gasket. It is great because you don't have to know anything to start making pretty pictures, but you immediately get a sense that there is something significant going on. For those who want to see beyond the pretty pictures, one could explain the contraction mapping theorem (terrific fun in its own right) and develop affine-linear transformations starting with rotations, scalings and translations. Trying to find the transformations which define a particular fractal by "finding enough smaller copies of the fractal to cover the fractal" is also great fun, which is compelling even for children.

In such a museum, I would like to see how mathematics are used in real life, not just for their internal beauty (well, beauty, simplicity and usability are certainly related). I mentioned above in a comment how Thales' theorem has been the tool to measure the height of pyramids. This can make a nice mathematics experiment: a lamp (the sun) a small pyramid and a stick. And suddenly math comes alive. Another kind of living mathematics: put salt into a thin aquarium such that the density vary, top to bottom, from zero to (almost) infinity. Send a light beam to the aquarium and the light will follow a geodesic of Poincaré's half plane (this experiment has been actually presented at the Paris "Palais de la découverte"). These are just two examples of "math in real life", I'm confident in mathematician's skills to find a lot more of such examples (not just in geometry: prime number and securing communications, statistics and controlling epidemics, etc...). I'm sure that understanding with our eyes how mathematics are used in real life makes mathematics even more sexy.

What about some large-number phenomena? This seems to be something the general public would appreciate and could relate to the "Computers in Modern Mathematics" booth others have suggested.

What I have in mind is not really Ackerman function/Graham's number business (which I don't think I could wrap my head around any more easily at a museum), but facts that involve small-ish large numbers. For instance:

The smallest positive integer $n$ for which $n$ divides $2^n-3$ is $4,700,063,447$.

There are many other great examples (though not all interesting or accessible to non-mathematicians) in answers to this MO question. It also might be nice to see comparisons of smallest counterexamples like this to 'real-world' numbers like the population of China (~$1.34$ billion), or the number of cells in the human body (~$10^{14}$), or the number of elementary particles in the observable universe (~$10^{80(\pm10?)}$).

To me, the goal of such an exhibit should be (1) to provide a few examples (like the one above) illustrating the importance of proof over verification of the first $10^{10}$ cases, and (2) to help museum-goers conceptualize the small-ish large numbers that come up in analyzing real-world phenomena.

In an Italian museum (probably the Leonardo da Vinci Museum in Florence) I saw a compass for drawing arbitrary conical sections. I believe the legend mentioned only ellipses, but it could in principle draw the others too.

The basic principle is that the "central" arm (in general, the focal arm) of the compass is held at a fixed (per drawing) angle to the desk, while the pencil arm adjusts in length (so that it is shorter at the perigee and longer at the apogee).

A room dedicated to waves, waterwaves, soundwaves and lightwaves illustrating interference, refraction, Fourier transform and so on with the help of concrete (and playful)
devices, and explaining that waves are as much mathematics (trigonometric functions,
differential equation, complex numbers) as physics (optics, acoustics, quantum mechanics).